I'm not sure there's anything iLogic (or any other customization) can bring to the table because there's not much in the API to call when it comes to the analysis tools.
How would I go about looping through and printing each combination of dimension parameter values and design constraint (FEA max displacement in above table) results?
To loop through the configurations, you can get it to generate the model for each parametric value:

Once you've done that you can drag the slider(s) to see the model change. Right-click to get more options - like studying a particular configuration.
Running the analysis gives you three options:

These are defined in the help as:
- Exhaustive set of configurations Solves all combinations of the parameters.
- Smart set of configurations Solves the base configuration and each parameter change to the base, then interpolates the other results.
- Current configuration only Solves the current configuration as a single point simulation.
If you want to refine your analysis, you can set "Limits" on the outputs you are analysing.
This shows I'm within my goals:

And this shows I'm not:

But that still doesn't let you take all the parametric options and their results out to a table. You can run a report to show all the outputs - but I don't think that's what you're looking for.
The output you are looking for would be complicated to display (even if it existed). You'd probably have to create a new sheet in Excel for each study - and for the model in your screenshot, that would be 122 sheets (11 x 11 + 1)
You might be better off running the exhaustive analysis - setting some limits and checking which parameter combinations are worthy of further investigation. Promote those configuration(s) back to the model and export the parameter(s). While not the answer you asked for, it hopefully gets you closer to the goal more quickly - which I guess is what parametric analysis is for.
Hope this helps.
Peter